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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Variational upper bound on the scattering length

Donald Gelman

Published Feb 15, 1974• 2 citations
Publisher

Abstract

Previous work on the application to scattering theory of the Feynman path-integral formulation of nonrelativistic quantum mechanics is extended. By using the regularization $\mathrm{Tr}({e}^{\ensuremath{-}\ensuremath{\beta}H}\ensuremath{-}{e}^{\ensuremath{-}\ensuremath{\beta}{H}^{0}})$ of the divergent partition function $\mathrm{Tr}{e}^{\ensuremath{-}\ensuremath{\beta}H}$, where ${H}^{0}$ is the free-particle Hamiltonian, $H={H}^{0}+V$, and $V$ is a short-range potential of the form $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ which cannot support a bound state, we obtain a variational upper bound on the scattering length. This bound is expressed in terms of path integrals containing a trial functional $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$, and reduces to the natural analog of the Rayleigh-Ritz result when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ is of the standard form $\ensuremath{-}\ensuremath{\int}\frac{[{E}_{K}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}]dt}{\ensuremath{\hbar}}$, with ${E}_{K}$ the kinetic energy and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ a trial potential. Since $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ need not be of this form, the bound contains new features.

DOI
10.1103/physrevd.9.931

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